While a wireless repeater would work as proposed by @David, if you are willing to run cables through the attic, you could get significantly better performance by using a cabled solution - and it will also be simpler, cheaper and more robust solution.
The solution would be to get an Access Point (You can get a router with WIFI and just ignore the WAN part), and connect an Ethernet cable from your Verizon connection to the Access Point.
If the Access Point is centrally located and can get good coverage throughout the house, it could also be beneficial to turn off the WIFI functionality on the Verizon connection [failing which, remove the aerial if that can be done]. Alternatively, locate the Access Point at the other extreme of the house, and ensure you select a different frequency. This could give you better coverage through the house, but spams more noise at your neighbours.
I note that you say the distance from the Verizon router to the center of the house is 75-100 feet. By standards around here, that implies you have a HUGE house, and you may be better off leaving the Verizon router on, and locating the other router 2/3rds to 3/4 of the way to the far other side of the house rather then centrally.
To help with the terminology (I'm simplifying just a bit) -
Switch - A switch, everything "sees" everything else. In general terms, the switch does not look at or filter any traffic.
Access Point - Converts between Ethernet and WIFI, but acts more-or-less like a switch (ie it does not look at or filter the internet connection, except as minimally required for WIFI security which is added)
A router looks at packets going between the interfaces and can rewrite/block/monitor them). "Commercial" routers can push traffic between many different links. Most home routers actually only have 2 "real" interfaces, the WAN Interface which connects to the ISP and the "LAN" ports which act like a [4 port] switch.
A bridge is a term you should not need to worry about as a small/home user. (When I said the LAN ports on a home router act like a switch, those ports are usually actually individual ports which can be handled differently by the hardware, but the hardware performs bridging [ is acts as a bridge ] to make them appear as a switch. Similarly, when you use an Access Point (or home router), the Ethernet and WIFI connection are bridged together as its much easier for people who are not network administrators to work with, as things "just work".
As far as cabling goes, it should not matter too much if its Cat 5, Cat5e, Cat 6 or Cat 6e or better - particularly if your Verizon connection is < 100 megs. That said, the difference in price between cat5 and Cat6 should be minimal, and if you don't have any lying around, go for Cat6. (Cat 5 < Cat5e < Cat 6 < Cat 6e). Of bigger concern then the rating might be whether its stranded or single core - again either will work, but stranded is better if it needs to bend, like a network lead, and single core is better if it will be fixed in position. If its an option though, you may be better off getting a pre-made cable as it will invariably outperform something you crimp yourself - particularly if you are not expert at it. (Higher rated cable will perform better at longer distances and faster speeds - rule of thumb Cat 5 for maximum speed of 100 megs, Cat 6 for Gigabit, Cat 6e for up to 10 gigabit)