DC step-up regulator wiring
Some observations:


The minimum voltage for the SN754410 (according to a TI datasheet) is 4.5V. You're using the driver outside its normal range if you're feeding it from a 3.7V LiPo.
(Gross simplification) Best-case output current for the 754410 is 1A. Assuming perfect 3.7V, 1A output from the driver (which it won't be), that's a theoretical maximum of 3.7W. Assuming your little booster is 80% efficient (which it won't be) that's 2.96W available for your load, or around 246mA for the solenoid. Again, this is assuming lots of ideals - reality will be worse. You didn't specify the DC resistance of the solenoid control winding so I cannot predict how much current it will draw. (\Gross simplification)
The regulator should have \$ 33 \mu F \$ of capacitance on its input per the web link you cited. It's not shown on your drawing.
You haven't mentioned the A-h capacity of your 3.7V LiPo. The Arduino will be eating some of that energy too. A heavy load will likely discharge it fairly quickly - if the boost continues to receive input, it will try and draw more and more current to keep the output regulated (a switching converter shows a negative impedance load to the source.)