OK, here's my suggestion. The circuit is powered from 12V DC and drives a relay whose contacts can switch the 110V AC to the fan directly.
The circuit is based around two ICs from the CD4000 CMOS family. U1 is a CD4093B which contains four identical "gates" called NAND gates with Schmitt trigger inputs. They are shown as the four D-shapes in the diagram, with two inputs on the left and one output on the right.
One input of each gate is tied to VDD. This makes each gate operate as an inverter with a Schmitt trigger input. An inverter simply drives its output to the opposite state to its input: if the input is low, it drives its output high, and vice versa.
I won't get into details in this description as I've explained it all several times already - if you're having trouble getting to sleep, read my tutorial posts at
https://www.electronicspoint.com/threads/led-flasher-help.263649/page-2#post-1580565 and
https://www.electronicspoint.com/threads/led-flasher-help.263649/page-2#post-1580565. Only a few parts are directly relevant but you can learn a lot from them.
U2 is a CD4024B binary counter/divider IC and it provides the two minute timer for the pump control signal.
Your flow switch connects to CN1. It must provide a closed circuit when there is flow, and an open circuit when there is no flow.
Normally, R1 pulls the voltage at its bottom end up to VDD, the positive supply rail. This is considered a logic "high" at pin 2 of U1. When the flow switch closes, it pulls that voltage down to 0V, which is considered a logic "low". R2 protects U1A's input from possible voltage spikes from interference etc.
So, U1A's output (pin 3) goes high when there's flow in the pipe. When this happens, C3 is allowed to charge up through R3. (When pin 3 is low, D1 holds C3 discharged.) If the flow sensor remains active steadily for about two seconds, C3's voltage will reach the rising input threshold voltage of U1B (on pin 6) and U1B's output will go low.
When U1B's output goes low, the RC pulse generator made from C2 and R5 will cause a brief low pulse on U1D's input (pin 12) and its output will go briefly high. This resets U2 and starts a pump run cycle.
U1C forms a Schmitt trigger oscillator (see the tutorial posts I linked to earlier) that oscillates (alternates high and low at a regular rate). The time period of one cycle is about 1.875 seconds (it can be set fairly accurately by VR1).
This frequency is used as a clock for U2, which is a 7-bit binary counter. If you're familiar with binary numbering, this IC generates a 7-bit binary number that increments (increases by 1) on each clock pulse.
The Q1~Q7 outputs are the binary number. When U2 is reset (which happens whenever the flow switch changes from OFF to ON and remains ON steadily for two seconds), the outputs reset to 0000000 binary. As the clock pulses come along, the outputs increment, to 00000001, then 0000010, then 0000011, and so on. After 64 clock pulses, the count reaches 1000000, i.e. the Q7 output (pin 3) goes high. This occurs after (1.875 × 64) = 120 seconds.
The Q7 output is named -PUMPRUN. This means that when it is low, the pump runs. (the "-" at the start of the name shows that it's an active low signal.)
When -PUMPRUN goes high, two things happen. First, D2 conducts and forces U1 pin 8 high. This disables the oscillator, so U2 remains stuck at that count. Second, the voltage to the base of Q1 goes high, and this causes the relay to open.
Q1 is used as an emitter follower. The voltage at its emitter follows its base voltage, with a slight voltage difference of about 0.7V which is not significant. When Q1's base is low, it pulls its emitter low as well and this applies most of the 12V power supply across the coil of relay K1, making it close and activate the pump. When Q1's base is high, the relay drops out.
Normally a diode would be connected across the relay coil to absorb the "inductive kickback" from the relay coil and prevent damage to Q1, but with Q1 used as an emitter follower, that diode isn't needed.
So when the circuit is idle, U2 contains a count of 1000000 binary; the Q7 output (pin 3) (-PUMPRUN) is high. The relay is de-energised, so the pump doesn't run, and D2 prevents U1C from oscillating.
When flow starts, and remains active for two seconds continuously, a short high pulse resets U2 to 0000000 binary. This activates the relay (because -PUMPRUN is now low), and enables the oscillator, and 2 starts counting up towards 10000000 binary.
I hope this makes sense. Try to understand those tutorial posts I pointed you to. This circuit is different in the details but a lot of the terminology and principles are relevant.
C1 and C5 are decoupling capacitors for U1 and U2 and must be connected as directly as possible to pins 14 and 7 of their respective ICs. Use ceramic capacitors for these, and for C2 as well (it's the same value).
I have given specific Digi-Key part numbers for C3 and C4, as well as two options for K1. Other components are not critical and can be by searching for their part numbers, or through the product guide.
Stripboard is an appropriate construction method for this circuit. The relay contacts are at mains voltage and must be thoroughly insulated from the other circuitry - remove a large area of copper around these to ensure good clearance and creepage distances.
VR1 should be set roughly half way. If this produces a pump run that's too short, turn it clockwise and try again. If the pump runs too long, turn it counterclockwise. The clockwise and counterclockwise ends of the trimpot are marked on the schematic.
